For $d$ odd these two components are disconnected, because $I$ and $-I$ induce opposite orientation. For $d$ even, you have an involution, which takes a lattice to a complex conjugate lattice. This involution takes a lattice
$\tau_1, ..., \tau_{2d}$ to a complex conjugate lattice 
$\bar\tau_1, ..., \bar\tau_{2d}$, and has many fixed points.

However, be very careful about this:
"By this https://mathoverflow.net/q/126329 the moduli space of complex $d$-tori is $X = GL_{d}(\mathbb{C})\backslash GL_{2d}(\mathbb{R})/GL_{2d}(\mathbb{Z})$": for $d\geq 2$, this double quotient is very non-Hausdorff,
because $\mathbb{R})/GL_{2d}(\mathbb{Z})$ acts on $GL_{d}(\mathbb{C})\backslash GL_{2d}$ with dense orbits, as follows from Ratner theory.
It's not very good idea to consider this quotient as an orbifold, because it has codiscrete topology: all open sets are either empty or everything.
Detailed explanation is found in this paper: https://arxiv.org/abs/1306.1498 ("Ergodic complex structures on hyperkahler manifolds")